This section of our PERT Math Study Guide covers how to interpret data sets. measures of center and probability are key concepts on the PERT and can present some tricky questions for test takers who don’t read carefully. Use our study guide below to review and practice your data interpretation skills.
Mean
The mean (or average) is calculated by adding all the values in a data set and dividing by the total number of values. It provides a measure of the overall level of the data, but it can be affected by extreme values (outliers).
Example: Find the mean of the following data set: 4, 8, 6, 10, 2
$\text{Mean} = \dfrac{(4 + 8 + 6 + 10 + 2)}{5} $ $ = \dfrac{30}{5} = 6$
Median
The median is the middle value when the data is ordered from smallest to largest. If the number of data points is even, the median is the average of the two middle values. The median is less affected by outliers than the mean, making it a better measure of central tendency in skewed distributions.
Example: Find the median of the following data set: 3, 7, 8, 12, 15
The median value is the middle value, which is 8.
Example: Find the median of the following data set: 3, 7, 10, 12
Here, there are an even number of values, so no number is right in the middle. We must average the middle two values.
$\text{Median} = \dfrac{(7 + 10)}{2} = 8.5$
Remember to put the data in numerical order from least to greatest, if it is not already in order, before finding the median.
Mode
The mode is the value that occurs most frequently in the data set. There can be more than one mode if multiple values have the same frequency. The mode is particularly useful for categorical data, where calculating an average is not meaningful.
Example: Find the moe of the following data set: 2, 4, 4, 6, 8, 8, 8, 10
The mode is 8 because this number appears most often (3 times).
When to Use Mean, Median, or Mode
Measures of central tendency (mean, median, and mode) describe the center of a data set. Knowing when to use each is key to interpreting data accurately.
- Mean is best for data without extreme values (e.g. test scores in a balanced class).
- Median is best for skewed data (e.g. incomes where a few high earners distort the average).
- Mode is best for identifying the most common result (e.g. shoe sizes in a store).
Basic Probability
Probability measures the likelihood of an event occurring and can be expressed as a fraction, decimal, or percentage. It is calculated using the following formula:
$\text{Probability} = \dfrac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}$
- Probability values range from 0 to 1, where 0 means the event is impossible and 1 means it is certain.
- Probabilities can also be written as percentages. For example, a probability of 0.25 is equivalent to 25%.
Example: When rolling a six-sided die, what is the probability of rolling a 3?
There are six possible outcomes (the six sides of the die) but only one favorable outcome (rolling a 3):
$\text{Probability} = \dfrac{1}{6}$
For compound events (e.g. rolling two dice), we must multiply the probabilities of independent events.
Example: When rolling two dice, what is the probability of rolling two sixes?
$\text{Probability} = \dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{36}$
Key Tip: Watch out for extreme values. Outliers can heavily influence the mean, but they do not have the same effect on median or mode. Be cautious when using the mean with data sets that contain extreme values.